Let \(X=\Gamma \backslash D\) be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain \(D=G(\mathcal{R})/V\) by a discrete subgroup \(\Gamma\). Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety \(Y \subset X\) (of Shimura type), and formulate necessary criteria for \(Y\) to be special. Our method consists in looking at finitely many compactified special curves \(C_i\) in \(Y\), and testing whether the inclusion \(\bigcup_i C_i \subset Y\) satisfies certain properties. One of them is the so-called relative proportionality condition. In this paper, we give a new formulation of this numerical criterion in the case of Mumford-Tate varieties \(X\). In this way, we give necessary and sufficient criteria for a subvariety \(Y\) of \(X\) to be a special subvariety of Shimura type in the sense of the André-Oort conjecture. We discuss in detail the important case where \(X=A_g\), the moduli space of principally polarized abelian varieties.

14G35

André-Oort conjecture, period domain, Shimura variety, Higgs bundle

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Universit\``{a}t Mainz, Fachbereich 08, Institut f\''ur Mathematik, 55099 Mainz, Germany

Universit\``{a}t Mainz, Fachbereich 08, Institut f\''ur Mathematik, 55099 Mainz, Germany

Universit\``{a}t Mainz, Fachbereich 08, Institut f\''ur Mathematik, 55099 Mainz, Germany